# Noether normalisation over $\mathbf{Z}$

Is there a Noether normalisation lemma for finitely generated (flat) algebras over $\mathbf{Z}$ (or more generally principal ideal domains)? It seems one can tensorise with the quotient field and then apply the usual Noether normalisation lemma. I couldn't find this in the literature, so I suspect it is wrong.

• – Asvin Oct 20 '17 at 17:14
• Isn't this the same question? math.stackexchange.com/questions/213336/… – Asvin Oct 20 '17 at 17:16
• @Asvin: Thanks! Can one omit the localisation in the case of a PID and a flat algebra? – user19475 Oct 20 '17 at 17:33
• I haven't actually gone through the links myself. I just remembered seeing similar questions before. Maybe you will find the formulation here more useful? mathoverflow.net/a/60716/58001 – Asvin Oct 20 '17 at 17:47
• @TimoKeller No, just look at the counterexample from the second question, with $\mathbb Z[1/2]$. – Will Sawin Oct 20 '17 at 19:17